Modelling of air flow rate in significantly flattened rounded rectangular ventilation ducts

Paper presents new mathematical model for air flow velocity distribution in rounded rectangular ducts and its experimental verification. In papers [1, 2] an mathematical model based on modified Prandtl equation for power power-law velocity profile was determined. It works very well for smaller cross sections. During the study of larger cross sections new phenomena in flowing air have been observed, it forced the search for a new model. The new model is based on a rounded rectangular division into two parts: slot and rounded square.


Introduction
An important element in the design of ventilation systems (first of all connected with air conditioning) is the selection of appropriate ducts. So far ducts of circular or rectangular shape are commonly used, sometimes of oval cross section area. For some time a new ventilation system has been developed, in which channels have a rounded rectangular cross section area.
This new system combines the advantages of circular and rectangular channels. It provides as well as rectangular ducts good use of space in rooms, allows good cleansing of them (no sharp corners), and has good flow properties. Although the theory of air flow in ducts is reasonably well understood, theoretical solutions are obtained only for a few simple cases such as fully developed laminar flow in a circular pipe [3]. Therefore, we must rely on experimental results and empirical relations for most air flows in ducts rather than closed-form analytical solutions. Noting that the experimental results of ventilation ducts are obtained under carefully controlled laboratory conditions and that no two systems are exactly alike, we must not be so naive as to view the results obtained as "exact." An error of 10 percent (or more) in friction factors calculated using values presented in different design manuals is the "norm" rather than the "exception." On the other hand, mathematical modelling of the air flow distribution in cross-sectional area of the duct is necessary. In Fig. 1, all dimension series of the developed new system ducts are shown, 79 cells are filled in this table. For each dimension there are about 40 different elements, this means that more than 3,000 items should be tested. This is not possible from a time point of view.
For this reason, it is necessary to develop a certain empirical model of air flow in cross section based on experimental research. The first step in the development of this model was the adoption of dimensionless parameters for characterization of the duct cross-section shape  and  , see  Cross sectional area is characterized by three parameters: width W , height H , and rounding radius R (Fig. 2). These parameters, during the analysis, are used in two non-dimensional forms: flatness of rectangle  and relative rounding  , where width W is the basic parameter.
The cross-sectional area of ducts varies from 2 _ min 0.051m . This wide scope of areas imposes high demands relating to the measurement stand. Rounding radius R is substantial for the degree of the rounded rectangular reduction area with respect to the rectangular cross section. In Fig. 3 impact of the rounding radius on the size of rounded rectangular the cross sectional area related to full rectangular cross section. This effect was determined by the formula: where: rr A -rounded rectangular area, rec A -full rectangular area (without rounded corners) For technological reasons only one rectangle rounding radius 0.1m R  was tested.

Modelling of volume flow rate
Modelling of volume flow rate is concerned mainly to rectangular cross sections for Therefore, a new model of velocity distribution in the cross section was to be found. The analysis below is only case-sensitive 0.5   , because the cross-sections for are only rotate the section by angle 90°, what is irrelevant from the mathematical point of view. It was noted that within a wide range of coordinates around the central point C , the local flow velocity was not changed. This suggested a presumption that velocity distribution in this area is similar to the velocity distribution in a slot. This observation confirms the well-known fact that the Reynolds number based on the hydraulic diameter h D for cross sections different from circular better fit the similarity of flows, if the cross section is closer to the circle., e.g. square, hexagon and so on. Therefore, the new model was taken over that the flow in the channel is twofold: one-dimensional as in slot and twodimensional as in circular cross-section (Fig 2).
Cross-section marked in the figure Fig. 2 side parts of the duct cross section together they form a square with rounded corners of side H , marked in the figure Fig. 2

Fig. 4 Idea of the rectangular slot
The volume flow rate through the elementary cross section is defined by the formula: When the integration of flow rate over the entire surface we obtain: Based on empirical studies, it can be assumed that for large sections the model resulting from the modified Prandtl formula is superior to that of its original model. According to the modified Prandtl formula (power-law velocity profile), the flow velocity distribution along the coordinate is defined by the formula [1]   After substituting (4) to (3) we obtain   The integral defined by the formula (5)  The integral expression in formula (7) is a beta function 1 Finally, we can write or using non-dimensional parameter  of cross sectionnal area Virtual both parts can be connected to a rounded square duct with centre of symmetry rsq C . Note that

Volume flow rate through the rounded square
This duct of cross section rsq A perimeter is created by side walls where the air velocity is zero (Fig. 5) Above mentioned equation (12)

Volume flow rate through the rounded rectangular
Volume flow rate through the rounded rectangular is sum of flow rates through slot and rounded square rr slot rsq

Measurement results
Received model that allows to determine volume flow rate through the rounded rectangular rr V  verified by measurements on a test bench. Test bench consist of flowmeter "testo", fan, tubular air deflector [4], and ventilation duct of 20 meters W  long. Initial part of test bench for measurement volume flow rate rr V  is presented in Fig. 6. Flowmeter "testo" was mounted on the fan inlet. This solution allows to use this measurement device for many measurements. Of course, it is assumed that the amount of incoming air is equal to the amount of exhaust air from duct line. . It has own tubular defector to calm the input flow. The fan rotational speed was controlled by the inverter. All measurements were provided near the end of duct. In Fig.7 is presented traversing system (left) and HWA probe placed 6 mm before the end of duct.  Presented in Fig. 8

Conclusions
The new model is very useful especially when modelling large significantly flattened rounded rectangular ventilation ducts. In this case the models based on Prandtl equation for power power-law velocity fail. They needs big number n , e.g. 20 n  . Unfortunately this model is more complicated. The main disadvantage is the gamma function, which requires application of advanced computational technique. However, in some applications it can be very useful, for example, developing a new type of flow meter, based on flow velocity measurement at the central point C.
This new model can also be used for smaller cross sectional, it is more universal.
The required measuring range of volume flow rate 3 1 0.257 11.957 m s V     exceeded the measuring range of the flowmeter. Therefore, the new model developed was used to determine the value of volume flow rate V  . This was necessary for the 57 ducts examined. Only for 22 cross sections it was possible use flowmeter "testo", therefore the new developed model has greatly helped in carrying out research.